Two rods A and B of different materials are welded together as shown in figure. Their thermal conductivities are $K_{1}$ and $K_{2}$ . The thermal conductivity of the composite rod will be

\frac{3 (K_{1} + K_{2})}{2}

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K_{1} + K_{2}

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2 (K_{1} + K_{2})

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\frac{(K_{1} + K_{2})}{2}

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Solution

The correct option is D $\frac{(K_{1} + K_{2})}{2}$
From the diagram we can observe that both the rods are in parallel, then the total heat transfer will we be the sum of heat transfer from both the rods i.e.,
$H_{t o t a l} = H_{A} + H_{B}$
As we know, $H = K A \frac{Δ t}{d}$
$K_{t o t a l} A_{t o t a l} \frac{Δ t}{d} = K_{1} A_{1} \frac{Δ t}{d} + K_{2} A_{2} \frac{Δ t}{d}$
Assuming, $A_{1} = A_{2} = A$
then, $A_{t o t a l} = A_{1} + A_{2} = 2 A$
$K_{t o t a l} 2 A = K_{1} A + K_{2} A$
$K_{t o t a l} = \frac{K_{1} + K_{2}}{2}$

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